How large does the photovoltaic system need to be?

If you are not just installing a church steeple roof, yes.

A pure photovoltaic system with 10 kWp currently costs 1400,-€/kWp (excluding tax).
20 kWp then only 1200,-€/kWp.
If it is more expensive, the solar installer does not want the contract. For whatever reasons.
A year ago, both were still about 200,-€ cheaper.

How much one wants to spend on a storage system beyond that (the ranges are huge here) can be decided individually. If it costs more than 250,- €/kW, it will probably not break even. It is useless in any case.

(All values are average values. There are the usual regional differences.)

If it costs more than €250/kW, it will probably not achieve the black zero.

Do you have a specific model that fits?

So I see that correctly? 10kwp with 6kwp storage including installation should be feasible for a maximum of €20,000 gross?

hm the current offers I get here in the North unfortunately say something different....

Yes, currently you have to search a bit because almost everyone is overloaded and therefore makes defensive offers.

Do you have a specific model that fits?

None is on the market yet. Unless self-construction is an option.

hm the current offers I get here in the North unfortunately say something else....

May I ask what was offered there?

10kWp, 18k gross. Storage 8kWh 4.5k gross Maybe worth mentioning that the 10kWp is equipped with individual power optimizers. This may somewhat justify the price.